Second derivative of solution from ode45

I have a system of 2 second order ODEs, which i converted to a system of fist order ODEs, which ode45 (or ode15s) needs as an input

my system input looks like this:

as an output i get the vector , but I need the derivatives and . How can i get those?

I tried using the gradient function as suggested by Star Strider in this question: How to output derivative from ODE 45 but unfortunately that did not help.

Thanks in advance!

Answer 1

[Bjorn Gustavsson]

After integrating your ODE-system you will have for some , and you have the function for your ODE-equations, f(t,X), one way to calculate the accelerations at times will be to simply call:

dxd2xdyd2ydt(i,:) = f(t_i,X(i,:));

Typically I use a regularly space array for the time in the call to ode45 and use the gradient-function, but the above should work - that is after all what we expect odeNN: to give us a solution X where that is true...

HTH

[e_frog]

ty for your answer! I dont quite understand what you mean by "function for your ODE- equations" or what the f would be. Could you please elaborate give me simple example?

[James Tursa]

You have expressions for ODE1 and ODE2. Simply plug the solution from ode45 into those expressions to get your 2nd derivative values.

[Bjorn Gustavsson]

@e_frog: Do you wonder how to write your ODE-function?

If so, let's make a 1-D harmonic oscillating ODE, i.e.

That would become a function something like this:

function dxdtd2xdt2 = ode_ex_ho(t,x,k)
dxdtd2xdt2 = [x(2);
              -k*x(1)];
end

Such a function you can adjust as you need.

HTH

[e_frog]

@Bjorn

Thats exactly how I converted my second order ODEs to a first order system. I would call the example function like this:

[t,X] = ode45(@(t,X) ode_exo_ho(t,X,k),tspan,y0)

The Problem with that is, that the coutput vector X

would in this example be

but I need

(respectively for my case with 2 ODEs I would need and )

[James Tursa]

Please show us the expressions for ODE1 and ODE2. I still don't understand why you are having trouble plugging x and xdot into these expressions to get xdotdot. Why can't you just call ode_exo_ho( ) with the appropriate inputs to get what you want?

[e_frog]

@james

do you mean I can just insert the double vectors in the ODE functions and calculate the result? wow.. that would be too easy... :'D

[James Tursa]

If the ode_exo_ho( ) function is vectorized (meaning it will accept vectors and properly calculate a vector result), then yes it is just this easy. If ode_exo_ho( ) is not vectorized, then you would have to call it in a loop, but still pretty easy.

The ease of vectorizing your function depends on what it is doing. Sometimes it is just a simple matter of replacing the operators * and / and ^ with their element-wise counterparts .* and ./ and .^

[e_frog]

ty it worked!